(Reuters) - Private equity firm Apollo Global Management LLC (APO.N) is nearing a deal to acquire Fresh Market Inc TFM.O, a U.S. specialty grocery retailer, for more than $1.3 billion, people familiar with the matter said on Friday.

The deal would underscore Apollo's confidence that it can turn around companies in the highly competitive U.S. grocery sector. The New York-based buyout firm's past investments include Sprouts Farmers Market (SFM.O) and Smart & Final Stores Inc (SFS.N).

Apollo has prevailed in an auction for Fresh Market with a cash offer of $28.50 per share, and is now finalizing the terms of the deal, one of the people said.

An agreement could be announced as early as Monday, the people added, cautioning it had not yet been finalized and that it was still possible for the deal to be amended or fall apart at the last minute.

Apollo declined to comment, while Fresh Market representatives did not immediately respond to requests for comment.

Fresh Market said in October it had retained investment bank JP Morgan Chase & Co (JPM.N) to assist in a review of strategic options, including the potential sale of the company. Fresh Market's shares ended trading in New York on Friday at $22.98.

Reuters had earlier reported, citing sources, that the Greensboro, North Carolina-based company's founder and board chairman Ray Berry was considering taking the company private.

Berry's son-in-law, Michael Barry, who previously served as Fresh Market's chief financial officer, was also considering rolling his 6.4 percent stake in the deal, the sources said at the time.

Fresh Market has 183 stores, which it operates across 27 states. Its predominately southeastern locations include North Carolina, Florida and Georgia. Its Mid-Atlantic and Midwest locations include Connecticut, New York, Kansas, Illinois and Oklahoma.

Last year, Fresh Market hired Rick Anicetti as chief executive to help battle low margins and declining same-store sales. He had previously served for eight years as President and CEO of Food Lion LLC's grocery stores.

Reporting by Greg Roumeliotis and Mike Stone in New Yorked; Editing by Robert Birsel